Optimal Parameter-dependent Bounds for Kuramoto-sivashinsky-type Equations
نویسندگان
چکیده
We derive a priori estimates on the absorbing ball in L2 for the stabilized and destabilized Kuramoto-Sivashinsky (KS) equations, and for a sixth-order analog, the Nikolaevskiy equation, and in each case obtain bounds whose parameter dependence is demonstrably optimal. This is done by extending a Lyapunov function construction developed by Bronski and Gambill (Nonlinearity 19, 2023–2039 (2006)) to take into account the dependence on both large and small parameters in the system. In the case of the destabilized KS equation, the rigorous bound lim supt→∞ ‖u‖ ≤ KαL3/2 is sharp in both the large parameter α and the system size L. We also apply our methods to improve previous estimates on a nonlocal variant of the KS equation.
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